I could prove the following assertions:
Let P be a polynomial of degree n in $\Bbb R$[X]:
a) P is divisible by $(x-1)^{k+1}$ is equivalent to P' divisible by $(x-1)^k$ and P(1) = 0.
b) Let P(1) = 0. P is divisible by $(x-1)^{k+1}$ is equivalent to $P_1$ = nP - xP' is divisible by $(x-1)^k$
Now the text of the problem asks to deduce from a) and b) that:
c) P is divisible by $(x-1)^{k+1}$ is equivalent to:
$a_0$ + $a_1$ + ... + $a_n$ = 0
$a_1$ + 2$a_2$ + ...+ n$a_n$ = 0
....
$a_1$ + $2^k$$a_2$ + ...+ $n^k$$a_n$ = 0
I can see that the first 2 equalities in the above system come from the fact that P(1) = 0 and P'(1) = 0, but was not able to deduce the other equalities. Can someone provide me a hint? Thanks a lot.